Some results on ( p, k ) -extension of the hypergeometric functions

In this study, we investigate a new natural extension of hypergeometric functions with the two parameters p and k which is so called ( p, k ) -extended hypergeometric functions”. In particular, we introduce the ( p, k ) -extended Gauss and Kummer (or conﬂuent) hypergeometric functions. The basic properties of the ( p, k ) -extended Gauss and Kummer hypergeometric functions, including convergence properties, integral and derivative formulas, contiguous function relations and differential equations. Since the latter functions contain many of the familiar special functions as sub-cases, this extension is enriches theory of k-special functions.


Introduction
The motivation of the present work arises from: (A) The special functions are very important tools in the different areas of mathematical physics, astronomy, applied statistics, applied sciences and engineering which have engaged many researchers, we refer the reader to [1][2][3].
(B) The hypergeometric function belongs to an important class of special functions of the mathematical physics and chemistry [3] with a large number of applications in different branches of the quantum mechanics, electromagnetic field theory, probability theory, analytic number theory, data analysis, etc (see, for instance, [1,2,[4][5][6]). Indeed, various classes of extended special functions have been obtained as special cases for the hypergeometric functions. Traditionally, the hypergeometric function known as Gauss function is defined by which is absolutely and uniformly convergent if |v| < 1, divergent when |v| > 1, and is absolutely convergent when |v| = 1, if Re(δ 3 − δ 1 − δ 2 ) > 0, where δ 1 , δ 2 , δ 3 are complex parameters with is the Pochhammer symbol (or the shifted factorial) and Γ(.) is gamma function. The function in (1.1) satisfy the following differential equation Nowadays, numerous investigations, for example, in recent works of Srivastava et al. [7,8], Jana et al. [9,10], Agarwal et al. [11,12], Fuli et al. [13] and Abdalla [14,15] to introduce extensions and generalizations of the hypergeometric functions, defined by Euler type integrals, associated with properties and applications. In particular, Diaz and Pariguan [16] introduced the k-analogue of gamma, beta and hypergeometric functions and proved a number of their properties. Since that period, many different results concerning the k-hypergeometric function and related functions have been considered by many researchers, for instance, Agarwal et al. [17], Mubeen et al. [18][19][20], Rahman et al. [21], Chinra et al. [22], Korkmaz-Duzgun and Erkus-Duman [23], Nisar et al. [24], Li and Dong [25], Yilmaz et al. [26] and Yilmazer and Ali [27].
Motivated by some of these aforesaid studies of the k-hypergeometric functions and related functions, we introduce the (p, k)− extended Gauss and Kummer hypergeometric functions and their properties. Relevant connections of some of the discussed results here with those presented in earlier references are outlined.
The manuscript is organized as follows. In Section 2, we list some basic definitions and terminologies that are needed in the paper. In Section 3, we introduce the (p, k)-extended Gauss and Kummer (or confluent) hypergeometric functions and discuss their regions of convergence. In Section 4, we obtain integral and differentiation formulas of the (p, k)-extended Gauss and Kummer hypergeometric functions. In addition, contiguous function relations and differential equations connecting these functions are established in Section 4. Finally, we point out outlook and observations in Section 5.

Preliminaries
In this section, we give some basic definitions and terminologies which are used further in this manuscript.
Definition 2.1. [16,26] For k ∈ R + , the k-gamma function Γ k (u) is defined by where u ∈ C \ kZ − . We note that Γ k (u) → Γ(u), for k → 1, where Γ(u) is the classical Euler's gamma function and (u) m,k is the k-Pochhammer symbol given in the form the relation between the Γ k (u) and the gamma function Γ(u) follows easily that where Re(u) > 0 and Re(v) > 0.
Clearly, the case k = 1 in (2.3) reduces to the known beta function B(u, v), and the relation between the k-beta function B k (u, v) and the original beta function Definition 2.3. [16,26,27] Let k ∈ R + and s 1 , s 2 , η ∈ C and s 3 ∈ C \ Z − 0 , then k-Gauss hypergeometric function is defined in Proposition 2.1. [16,26] For any δ ∈ C and k ∈ R + , the following identity holds The k-hypergeometric differential equation of second order defined in [18,[25][26][27] by Particular choices of the parameters s 1 , s 2 , s 3 and k in the linearly independent solutions of the differential equation (2.6) yield more than 24 special cases. Also, the k-hypergeometric function can be given an integral representation in the following result [20,26]: Assume that η, s 1 , s 2 , s 3 ∈ C such that Re(s 3 ) > Re(s 2 ) > 0 and k ∈ R + , then the integral formula of the k-hypergeometric function is given by Furthermore, the k-Kummer (confluent) hypergeometric function 1 k 1 defined in [24] in the form The (p, k)-extended hypergeometric functions In this section, we introduce and discuss the (p, k)-extended Gauss hypergeometric function W(p, k; ξ) and (p, k)-extended Kummer (or confluent) hypergeometric function Y(p, k; ξ) as follows, respectively where k ∈ R + and ζ 1 , ζ 2 , ξ ∈ C and ζ 3 ∈ C \ Z − 0 , p is appositive integer and (ζ) m,k is the k-Pochhammer symbol defined in (2.2).
Remark 3.1. Some important special cases of the W(p, k; ξ) and the Y(p, k; ξ) for some particular choice of the parameters p and k are enumerated below: 1. Putting p = 1, we produce the k-analogue of Gauss and Kummer hypergeometric functions are given in (2.4) and (2.8), respectively.
2. Setting k = 1, we obtain a p-extension of the Gauss and Kummer hypergeometric functions in the following forms, respectively 3. Taking k = 1 and p = 1 in (3.1), we produce the standard Gauss hypergeometric function in (1.1).
The following theorem shows that the convergence property of the series (3.1).
Theorem 3.1. For all k ∈ R + and p > 1, then the (p, k)-extended Gauss hypergeometric function W(p, k; ξ) given by (3.1) is an entire function.
The following result can be verified in a similar way.  Remark 3.2. For p = 1 in Theorem 3.1, and Theorem 3.2, we get the convergence property of the k-Gauss hypergeometric function W(1, k; ξ) and the k-Kummer hypergeometric function Y(1, k; ξ), provided that k ∈ R + and ζ 3 ∈ C \ Z − 0 (see [16]). Remark 3.3. For p = 1 in Corollary 3.1, we obtain the convergence property of the usual Gauss and Kummer hypergeometric series (see [1,2]).

Integral representations and derivative formulae 4.1 Integral representations
Following, we establish the following theorems in terms of the k-integral representations of the (p, k)-extended Gauss and Kummer hypergeometric functions.
Proof. Considering the following elementary identity involving the k-Beta function B k (u, v) : in (3.1) and using the relation (3.4), we get the required integral formula (4.1).
Proof. Inserting the k-Pochhammer symbol (α 1 ) n,k from (2.2) in the definition (3.1) by its integral form given by (2.1) and from the relation (3.2), we thus obtain the desired result (4.2).

Derivative formulae
Theorem 4.4. The following derivative formulas hold true: Proof. The result (4.5) is obviously valid in the trivial case when n = 0. For n = 1, by the power series representation (3.1) of 2 F (p,k) 1 , we see from (4.5) that Replacing the k-pochhammer symbols (ζ 1 + k) m,k by the relation (2.2), we arrive at Therefore, the general result (4.5) can now be easily derived by using the principle of mathematical induction on n ∈ N 0 . A similar procedure yields the desired representation (4.6).
Proof. By using the series (3.1) in (4.7) and differentiating term by term under the sign of summation, we observe that which, in view of the series (3.1), yields the coveted formula (4.7). Similarly, we can derive the derivative formula (4.8).
Remark 4.3. The special cases of (4.7) and (4.8) when p = 1 are easily seen to reduce to the known derivative formulas of the k-Gauss and Kummer hypergeometric functions Remark 4.4. If we take p = 1 and k = 1 in the above mentioned theorems, we obtain the corresponding results for the classical hypergeometric functions 2 F (1,1) 1 and 1 F (1,1) 1 (cf. [6]).

Contiguous function relations and differential equations
The k-analogue of theta operator kΘ as given in [18,19,25], takes the form kΘ = k ξ dξ . This operator has the particularly pleasant property that kΘξ m = kmξ m , which makes it handy to be used on power series. In this section, relying on definition 2.1, we present some results concerning contiguous function relations and differential equations for the (p, k)-extended Gauss hypergeometric function 2 F To realize that, we increase or decrease one and more of the parameters of the (p, k)-extended Gauss hypergeometric function, W = W(p, k; ξ) = 2 F (p,k) 1 ζ 1 , ζ 2 ζ 3 ; ξ , k ∈ R + , p > 1, by ±k, then the resultant function is said to be contiguous to W(k; ξ). For simplicity, we use the following notations W(p, k; ζ 1 ±) = 2 F (p,k) 1 ζ 1 ± k, ζ 2 ζ 3 ; ξ .
From the above relations, we can easily obtain the following results